Sets And Relations Question 8
Question 8 - 01 February - Shift 2
Let $P(S)$ denote the power set of $S={1,2,3, \ldots, 10}$.
Define the relations $R_1$ and $R_2$ on $P(S)$ as $A R_1 B$ if $(A \cap B^{c}) \cup(B \cap A^{c})=\varnothing$ and $A R_2 B$ if $A \cup B^{c}=$ $B \cup A^{c}, \forall A, B \in P(S)$. Then :
(1) both $R_1$ and $R_2$ are equivalence relations
(2) only $R_1$ is an equivalence relation
(3) only $R_2$ is an equivalence relation
(4) both $R_1$ and $R _{\text{, are not equivalence relations }}$
Show Answer
Answer: (1)
Solution:
Formula: Equivalence relation (vii)
$S={1,2,3, \ldots . .10}$
$P(S)=$ power set of $S$
$AR, B \Rightarrow(A \cap {{}B^{c}}) \cup({{}A^{c}} \cap B)=\phi$
$R 1$ is reflexive, symmetric
For transitive
$(A \cap B^{c}) \cup(A^{c} \cap B)=\phi ;{a}=\phi={b} \quad A=B$
$(B \cap C^{c}) \cup(B^{c} \cap C)=\phi \therefore B=C$
$\therefore A=C$ equivalence.
$R_2 \equiv A \cup B^{c}=A^{c} \cup B$
$R_2 \to$ Reflexive, symmetric
for transitive
$A \cup B^{c}=A^{c} \cup B \Rightarrow{a, c, d}={b, c, d}$
${a}={b} \therefore A=B$
$B \cup C^{c}=B^{c} \cup C \Rightarrow B=C$
$\therefore A=C \quad \therefore A \cup C^{c}=A^{c} \cup C$
$\therefore$ Equivalence
